Conflict-Free Colourings of Graphs and Hypergraphs

Abstract

A colouring of the vertices of a hypergraphHis calledconflict-freeif each hyperedgeEofHcontains a vertex of ‘unique’ colour that does not get repeated inE. The smallest number of colours required for such a colouring is called theconflict-free chromatic numberofH, and is denoted by χCF(H). This parameter was first introduced by Even, Lotker, Ron and Smorodinsky (FOCS2002) in ageometricsetting, in connection with frequency assignment problems for cellular networks. Here we analyse this notion for general hypergraphs. It is shown that$\chi_{\rm CF}(H)\leq 1/2+\sqrt{2m+1/4}$, for every hypergraph withmedges, and that this bound is tight. Better bounds of the order ofm1/tlogmare proved under the assumption that the size of every edge ofHis at least 2t− 1, for somet≥ 3. Using Lovász's Local Lemma, the same result holds for hypergraphs in which the size of every edge is at least 2t− 1 and every edge intersects at mostmothers. We give efficient polynomial-time algorithms to obtain such colourings.Our machinery can also be applied to the hypergraphs induced by the neighbourhoods of the vertices of a graph. It turns out that in this case we need far fewer colours. For example, it is shown that the vertices of any graphGwith maximum degree Δ can be coloured with log2+εΔ colours, so that the neighbourhood of every vertex contains a point of ‘unique’ colour. We give an efficient deterministic algorithm to find such a colouring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need to (1) correct a small error in the Molloy–Reed approach, (2) restate and re-prove their result in adeterministicform.